library(M3C)
library(NMF) 
library(gplots)
library(ggsci)
# now we have loaded the mydata and desx objects (with the package automatically)
# mydata is the expression data for GBM
# desx is the annotation for this data

3.1 Exploratory data analysis

This is an important exploratory step prior to running M3C. It is best to remove extreme outliers. It is also important to be aware of the assumptions of PAM, K-means, and HC. K means, PAM, and HC assume the following;

1. Clusters are approximately spherical - not severely elongated in one direction (anisotropic) or non-linear
2. Clusters are approximately equal in variance

You can read more about the assumptions of clustering algorithms here: (http://scikit-learn.org/stable/modules/clustering.html)

Spectral clustering may be used to cluster more unusual structures using M3C, but this normally is not necessary. Ward’s hierarchical clustering (HC) is faster than these 3 algorithms and is an option included in M3C, although in practice PAM and KM usually perform better.

It is also important to check that batch effects and other uninteresting sources of variation do not drive the main variation in the data. The PCA function has a labels flag for adding categorical and continuous variables to the below plot (type ?pca). For a more quantitative analysis of these variables before clustering there is the excellent plot_drivers function from the bioplotr package which produces a very nice plot (https://github.com/dswatson/bioplotr).

The data should have been transformed to be homoskedastic (or nearly so) by this point (e.g. vst or log2), otherwise the larger features will have larger variances and dominate the PCA and cluster analysis. If the data contains different kinds of features, e.g. both protein and miRNA, then they should also be z-score normalised to be comparable.

PCA1 <- pca(mydata)

3.2 Running M3C

In our example, we run the algorithm using the default settings (100x Monte Carlo iterations and 100x inner replications). We have found the results generally stable using these parameters. If you are running a slow machine, have a massive dataset, or want results quickly, it is reasonable to lower the Monte Carlo iterations parameter to 50-5x. The inner replications needs to be at least 100x, changing this up to 250x will lead to increased result stability on some datasets.

Plots from the tool and an .csv file with the numerical outputs may be printed into the working directory (by adding printres = TRUE). We will set the seed in this example, incase you wish to repeat our results exactly (seed = 123). We will add an annotation file for streamlined downstream analyses (des = desx), and because we are comparing the classes with a categorical variable (tumour type) we shall set ‘doanalysis’ to TRUE and ‘analysistype’ to ‘chi’, to do a chi-squared test for significance. The ‘variable’ parameter is set to class (tumour histology in this data).

It is recommended to save the workspace after M3C if you are working with a large dataset because the runtimes can be quite long. M3C uses by default PAM with Euclidean distance in the consensus clustering loop because we have found this runs fast with good results. We will set the ‘removeplots’ parameter to TRUE in this example to remove plots from the vignette, normally this is FALSE by default.

res <- M3C(mydata, cores=1, seed = 123, des = desx, removeplots = TRUE, 
           analysistype = 'chi', doanalysis = TRUE, variable = 'class')

The scores and p values are contained within the res$scores object. We can see below the RCSI reaches a maxima at K = 4 (RSCI=0.33), the Monte Carlo p value supports this optimal K decision (p=0.033). This means the null hypothesis that K = 1 can be rejected for this dataset because we have achieved significance (alpha=0.05) versus a dataset with no clusters. For p values that extend beyond the lower limits imposed by the Monte Carlo simulation, M3C estimates parameters from the simulation to generate a beta distribution. The BETA_P in this case study is 0.033.

res$scores

##    K  PAC_REAL   PAC_REF        RCSI MONTECARLO_P     BETA_P   P_SCORE
## 1  2 0.6408163 0.5847347 -0.09158464   0.58415842 0.61114800 0.2138536
## 2  3 0.4473469 0.5490776  0.20490525   0.15841584 0.17714343 0.7516749
## 3  4 0.3526531 0.4762531  0.30046461   0.02970297 0.03541014 1.4508724
## 4  5 0.3191837 0.4086122  0.24699993   0.05940594 0.04699842 1.3279167
## 5  6 0.3061224 0.3492082  0.13168302   0.18811881 0.17563554 0.7553876
## 6  7 0.2946939 0.3043755  0.03232506   0.41584158 0.41553858 0.3813886
## 7  8 0.2751020 0.2708571 -0.01555055   0.58415842 0.55673522 0.2543513
## 8  9 0.2563265 0.2439429 -0.04951814   0.62376238 0.64921812 0.1876094
## 9 10 0.2244898 0.2216327 -0.01280896   0.49504950 0.54812339 0.2611217

Also important is the relationship between the clinical variables and the discovered clusters. In this data we want to compare with a categorical variable so perform a chi-squared test. We are reassured to see below K=4 is highly significant (p=5.5x10-14), however, K=5 is slightly more so. It is important to bear in mind the clinical or biological relationships as well as the structural nature of the data when deciding K.

res$clinicalres

##    K          chi
## 1  2 8.977979e-07
## 2  3 2.961138e-12
## 3  4 5.479839e-14
## 4  5 7.124742e-13
## 5  6 1.294080e-11
## 6  7          NaN
## 7  8          NaN
## 8  9          NaN
## 9 10          NaN

Now we will take a look at some of the plots M3C generates.

This is a CDF plot of the consensus matrices for our test data. We are looking for the value of K with the flattest curve and this can be quantified using the PAC metric (Senbabaoglu et al., 2014). In the CDF and following PAC plot we can see the inherant bias of consensus clustering where as K increases so does the apparent stability of the results for any given dataset, this we correct for by using a reference.

This figure below is the PAC score, we can see an elbow at K = 4 which is suggestive this is the best K. However, the bias of the PAC score can be seen here as it naturally tends towards lower values as K increases, thus making selecting K without taking this into account challenging to say the least. Furthermore, the PAC score cannot reject the null hypothesis K=1, for that we have introduced p values into the consensus clustering methodology.

We then derive the Relative Cluster Stability Index (RCSI) which takes into account the reference PAC scores using the reference mean. This metric is better than the PAC score for deciding class number, where the maximum value corresponds to the optimal K. In this example the RCSI has an optima at K=4. We recommend the RCSI be used to select K, and the p values to reject the null in most cases.

Finally, we calculate a p value from the distribution, here we display the p values from the beta distribution. If none of the p values reach significance over a reasonable range of K (e.g. 10), then we accept the null hypothesis. In the GBM dataset, we can see K = 4 reaches signfiicance with an alpha of 0.05 (red dotted line), therefore we can reject the null hypothesis K=1 for the GBM dataset.

Now we are pretty convinced there are 4 clusters within this dataset which are not likely simply to have occurred by chance alone.

A further analysis that M3C conducts (if the dend flag is set to TRUE) is to make a dendrogram for the optimal K using the consensus cluster medoids, then sigclust is run to calculate p values for each branchpoint. This allows quantification of the structural relationships between consensus clusters. In this case CC3 and CC4 have a closer relationship, whilst the other clusters are quite far apart.

We can turn to examine the output objects that M3C generates. These allow heatmap generation for publications.

3.3 Understanding M3C outputs

The first 3 lines below extract the ordered (according in clustering results) expression data and the ordered annotation data from the results object after running M3C for a 4 cluster solution. If we, for example, wanted to extract the data for a 5 cluster solution from the M3C results list, we would simply replace 4 in the below lines to 5. We then take a brief glance at the annotation object M3C outputs, a consensus cluster column has been added by M3C.

# get the data out of the results list (by using $ - dollar sign) for K=4
data <- res$realdataresults[[4]]$ordered_data # this is the data
annon <- res$realdataresults[[4]]$ordered_annotation # this is the annotation
ccmatrix <- res$realdataresults[[4]]$consensus_matrix # this is the consensus matrix
head(annon)

##                     consensuscluster     class
## TCGA.02.0048.01A.01                1 Proneural
## TCGA.08.0517.01A.01                1 Proneural
## TCGA.08.0350.01A.01                1 Proneural
## TCGA.08.0524.01A.01                1 Proneural
## TCGA.06.0648.01A.01                1 Proneural
## TCGA.02.0432.01A.02                1 Proneural

Next, we scale the data here row wise according to z-score, prior to some light data compression for visualisation purposes in the heatmap. Remember to set Colv = NA for heatmap plotting because the data columns (samples) are already ordered.

Note, we are going to use the NMF aheatmap function to draw the heatmap because it is simple, but for more complex heatmap designs for publication, ComplexHeatmap is definately preferable (https://bioconductor.org/packages/release/bioc/html/ComplexHeatmap.html).

data <- t(scale(t(data))) # z-score normalise each row
data <- apply(data, 2, function(x) ifelse(x > 4, 4, x)) # compress data 
data <- apply(data, 2, function(x) ifelse(x < -4, -4, x)) # compress data

ann_colors <- ggsci::pal_startrek("uniform")(4)
ann_colors2 <- ggsci::pal_futurama()(4) #
pal <- rev(colorRampPalette(RColorBrewer::brewer.pal(10, "RdBu"))(256))
NMF::aheatmap(data, annCol = annon, Colv = NA, distfun = 'pearson', 
         color = gplots::bluered(256), 
         annColors = list(class=ann_colors, consensuscluster=ann_colors2))

Another plot we may want to do for publications is print the consensus matrix for our optimal clustering solution (in this case, K = 4). This should be quite crisp reflecting the significant stability of the results. We can see in this heatmap below of the consensus matrix the clusters do indeed look quite clear supporting our view that there is 4 clusters.

Viewing consensus matrices manually should not be used to decide K, better to use the RCSI or p values that M3C provides to quantify the mixing proportions in the consensus matrix versus the K=1 null model. We also caution against the use of cluster validity metrics on the consensus matrices without first testing their behaviour in details on simulated positive control datasets. Our data indicates that these metrics may have substantial bias in them as well due to the underlying method.

# set up colours
n <- 10
seq = rev(seq(0,255,by=255/(n)))
palRGB = cbind(seq,seq,255)
mypal <-rgb(palRGB,maxColorValue=255)
# plot consensus matrix heatmap, do not cluster rows and columns
NMF::aheatmap(ccmatrix, 
              annCol = annon[,1,drop=FALSE], 
              color = mypal, scale = 'none', cexRow = 0, cexCol = 0,
              Colv=NA,Rowv=NA,annColors = list(consensuscluster=ann_colors2))

3.4 Final check of consensus cluster structure

A last analysis we recommend to do is to examine how the clusters are arranged in principal component and t-SNE latent variable space. To do this the pca and t-SNE functions can be run directly on the M3C output object. We may need to experiment with the perplexity parameter of t-SNE in the range of 5-20 to better suit the data. Both functions can also be used to print their results to the working directory. It is also possible to run UMAP on the data in a streamlined manner, see ?umap.

  PCA2 <- pca(res, K=4)
  TSNE <- tsne(res, K=4, perplex=15)

We can see in the PCA and t-SNE plots that clusters 3 and 4 are closer together. Doing a visual check like this allows the user to ensure there are no obvious artifacts that have occured.

4.1 Running M3C in fast mode

The second method minimises log(PAC) subject to lambda*K to find the optimal K. This metric is called the Penalised Cluster Stability Index (PCSI).

We recommend using this method if the user has a massive dataset or little time. It will correct for the overestimation of K observed with the PAC score.

res <- M3C(mydata, method = 2)

Here we can see the PCSI is minimised at K=4, this is the optimal K using this method.

5.1 Generating simulated data

We have included a function for generating simulated data to test various clustering algorithms. This cluster simulator is simple to use. Using the code below, clustersim generates a dataset with 225 samples, 900 features, a radius cut-off for the initial square of 8, a cluster number of 4, a seperation of clusters of 0.75, and a degree of noise added to each co-ordinate of 0.025. After running, a PCA will print of the data so we can visualise the 4 clusters in principle component space.

Note, clustersim’s method has recently been replaced with the CRAN package, clusterlab. We recommend using this for more sophisticated benchmarking. Clusterlab can generate Gaussian clusters with controlled variance and spacing, it can also create more complex higher order structures.

  res <- clustersim(225, 900, 8, 4, 0.75, 0.025, print = FALSE, seed=123)

5.2 Filtering features by variance

A simple feature filter function has been included in M3C that uses one of variance, median absolute deviation (MAD), Pearson’s coefficient of variation (A), or its second order derivative (A2) to filter the data. It is best to examine the distribution of variance with respect to the mean when selecting a filtering metric and cut-off setting.

MAD will yield similar results to variance, however, because it uses the median instead of the mean it will be more robust to outliers. This will be the preferable metric in many cases, although if the variance increases with the mean then A2 should be considered.

The statistics for each feature (and the filtered data itself) are saved in a convienient data frame in the results list for plotting and further analysis. In the first case we recommend to set percentile to 100 which collects statistics for all features. We then suggest to do some manual examination and plotting using the results of this function, for example, with the qplot function from ggplot2 of the mean-variance trend. If the variance increases with the mean, A or A2 may be more appropiate than variance or MAD.

Below, we set the percentile to filter by and input the data. The code below extracts the 10% most variable features using MAD as the filtering metric. In this case the example data has been already been filtered for variance then scaled additionally, so the results are not intended to be interpreteted as it is just an example. The statistics for the topN features will be printed to the console. Type ?featurefilter for more information.

  filtered_results <- featurefilter(mydata, percentile=10, method='MAD', topN=5)

## ***feature filter function***

## extracting the most variable: 10 percent

## features to start with: 1740

## performing calculations for median absolute deviation

## printing topN most variable features with statistics...

##        feature      mean      var       sd       MAD
## RPS4Y1  RPS4Y1 0.7868840 9.275727 3.045608 0.6429591
## POSTN    POSTN 0.2967294 7.301561 2.702140 3.2059149
## TMSL8    TMSL8 0.0846228 4.988073 2.233399 2.6527569
## LTF        LTF 0.0117520 4.580893 2.140302 3.0464465
## DKK1      DKK1 0.0621860 4.440944 2.107355 0.8973288

## features remaining: 174